Mathematics for Business Administration: Matrix Algebra

Mathematics for Business Administration:Matrix Algebra and Quadratic forms

Marıa Pilar Martınez-Garcıa



Marıa Pilar Martınez-Garcıa


Outline

Matrix Algebra

Matrices and Matrix OperationsDeterminants

Review problems for Matrix algebraMultiple choice questions

Quadratic forms

Definiteness of a quadratic formThe sign of a quadratic form attending the principal minorsQuadratic forms with linear constraints

Review problems for Quadratic formsMultiple choice questions

Useful Links


Matrix algebraReview Problems for Matrix algebra

Multiple choice questions

Matrix Algebra

Back

Matrix Algebra



Outline

Matrices and Matrix Operations

Determinants

Back

Matrix Algebra



Matrices and matrix operationsDeterminants

Definition of matrix

A matrix is simply a rectangular array of numbers considered as anentity. When there are m rows and n columns in the array, wehave an m-by-n matrix (written as m× n). In general, an m× nmatrix is of the form

A =

a11 a12 · · · a1na21 a22 · · · a2n

......

...am1 am2 · · · amn

,

or, equivalentlyA = (aij)m×n .

Matrix Algebra




Definition 1

A matrix with only one row is also called a row vector, and amatrix with only one column is called a column vector. We refer toboth types as vectors.

Definition 2

If m = n, then the matrix has the same number of columns asrows and it is called a square matrix of order n and is denoted byA = (aij)n.

Definition 3

If A = (aij)n is a square matrix, then the elements a11, a22, a33,..., ann constitute the main diagonal.

Matrix Algebra




Definition 4

The identity matrix of order n, denoted by In (or by I), is the n×nmatrix having ones along the main diagonal and zeros elsewhere:

I =

1 0 · · · 00 1 · · · 0...

......

0 0 · · · 1

Matrix Algebra




Addition of Matrices

A+B = (aij)m×n + (bij)m×n = (aij + bij)m×nRules

(A+B) + C = A+ (B + C)A+B = B +AA+ 0 = A

Matrix Algebra




Multiplication by a real number

αA = α (aij)m×n = (αaij)m×nRules

(α+ β)A = αA+ βAα(A+B) = αA+ βB

A+ (−A) = 0

Matrix Algebra




Matrix Multiplication

A ·B = (aij)m×n · (bij)n×p = (cij)m×p such that

cij =∑n

r=1 airbrj = ai1b1j + ai2b2j + · · ·+ ainbnj

Rules

(AB)C = A(BC)A(B + C) = AB +AC(A+B)C = AC +BC

Matrix Algebra




The transpose

A =

a11 a12 · · · a1n

a21 a22 · · · a2n

......

...am1 am2 · · · amn

⇒ A′ = At =

a11 a21 · · · am1

a12 a22 · · · am2

......

...a1n a2n · · · amn

Matrix Algebra




Symmetric matrix

Definition 5

Square matrices with the property that they are symmetric withrespect the main diagonal are called symmetric.

The matrix A = (aij)n is symmetric ⇔ aij = aji ∀i, j = 1, 2, ..., n.A matrix A is symmetric ⇔ A = A′.

Matrix Algebra




Determinants

Definition 6

Let A be an n× n matrix. Then det(A) or |A| is a sum of n!terms where:

1 Each term is the product of n elements of the matrix, withone element (and only one) from each row, and one (and onlyone) element from each column.

2 The sign of each term is + or − depending on whether thepermutation of row subindexes is of the same class as thepermutation of the column subindices or not.

Matrix Algebra




Determinants of order 2

A =

(a11 a12a21 a22

)⇔ |A| =

∣∣∣∣ a11 a12a21 a22

∣∣∣∣ = a11a22 − a12a21

Matrix Algebra




Determinants of order 3

A =

a11 a12 a13a21 a22 a23a31 a32 a33

|A| = a11a22a33 + a12a23a31 + a13a21a32− a13a22a31− a12a21a33− a11a23a32.

Matrix Algebra




Determinants of diagonal matrices

Diagonal matrix:∣∣∣∣∣∣∣∣∣

a11 0 · · · 00 a22 · · · 0...

......

0 0 · · · ann

∣∣∣∣∣∣∣∣∣ = a11a22 · · · ann

Matrix Algebra




Determinants of triangular matrices

upper triangular matrix:∣∣∣∣∣∣∣∣∣

a11 a12 · · · a1n0 a22 · · · a2n...

......

0 0 · · · ann

∣∣∣∣∣∣∣∣∣ = a11a22 · · · ann

The same occurs to lower triangular matrices.

Matrix Algebra




Rules for determinants

Let A be an n× n matrix. Then

|A| = |A′| .If all the elements in a row (or column) of A are 0 then|A| = 0.

If all the elements in a single row (or column) of A aremultiplied by a number α, the determinant is multiplied by α.

If two rows (or two columns) of A are interchanged, the signof the determinant changes, but the absolute value remainsunchanged.

Matrix Algebra






|A| = |A′| .

If all the elements in a row (or column) of A are 0 then|A| = 0.



Matrix Algebra









Matrix Algebra









Matrix Algebra









Matrix Algebra





If two rows (or columns) of A are equal or proportional, then|A| = 0.

The value of the determinant of A is unchanged if a multipleof one row (or column) is added to a different row (orcolumn) of A.

The determinant of the product of two matrices A and B isthe product of the determinants of each of the factors:

|AB| = |A| |B| .

Is α is a real number

|αA| = αn |A| .

Matrix Algebra








|AB| = |A| |B| .


|αA| = αn |A| .

Matrix Algebra








|AB| = |A| |B| .


|αA| = αn |A| .

Matrix Algebra








|AB| = |A| |B| .


|αA| = αn |A| .

Matrix Algebra




Practical methods to calculate |A|

In practice, there are two methods to calculate the determinants ofa square matrix (mainly used when its order is higher than 3):

Triangularization: Rule number 6 allows us to convert matrixA into one that is (upper or lower) triangular. Because thedeterminant will remain unchanged (as is stated by theproperty) its value will be equal to the product of theelements in the main diagonal of the triangular matrix.

Matrix Algebra





Expansion of |A| in terms of the elements of a row:

det(A) =n∑

j=1

(−1)i+jaij det(Aij)

= ai1(−1)i+1 det(Ai1) + ai2(−1)i+2 det(Ai2) + · · ·+ ain(−1)i+n det(Ain).

where ai1, ai2, · · · ain are the elements of the row i and Aij is the determinant

of order n− 1 which results from deleting row i and column j of matrix A (it is

called a minor).

Matrix Algebra





Expansion of |A| in terms of the elements of a column:

det(A) =

n∑i=1

(−1)i+jaij det(Aij)

= a1j(−1)1+j det(A1j) + a2j(−1)2+j det(A2j) + · · ·+ anj(−1)n+j det(Anj).

where a1j , a2j , · · · anj are the elements of the column j and Aij is the

determinant of order n− 1 which results from deleting row i and column j of

matrix A (it is called a minor).

Matrix Algebra




Definition 7

The product (−1)i+j det(Aij) is called the adjoint element of aij .

Matrix Algebra



Review Problems for Matrix algebra




Problem 1 Answer

Given the following matrices

A =

(1 11 1

)B =

(0 1−1 1

)C =

(−2 10 0

)confirm that the following properties are true

a (A+B)C = AC +BC.

b (AB)t = BtAt.

c (A−B)2 = A2 +B2 −AB −BA.

d (AB +A) = A(B + I), where I is the identity matrix of order2.

e (BA+A) = (B + I)A, where I is the identity matrix of order2.




Problem 2 Answer

Calculate AB and BA, where

a A =

(4 1 21 0 −5

)and B =

3 08 4−2 3

b A =

−1 2 02 0 11 1 1

and B =

2 3 10 2 41 5 3

c A =

(1 0 1

)and B =

1 23 05 7

Can we say that the product of matrices has the commutativeproperty?




Problem 3 Answer

Let A and B be square matrices. Prove that the property(A+B)2 = A2 +B2 + 2AB is false.




Problem 4 Answer

Calculate the following determinants:

a)

∣∣∣∣∣∣1 3 2−1 3 12 0 −3

∣∣∣∣∣∣ b)

∣∣∣∣∣∣∣∣3 1 −1 1−1 2 0 32 3 4 7−1 −1 2 −2

∣∣∣∣∣∣∣∣c)

∣∣∣∣∣∣∣∣3 1 2 −1−4 1 0 34 −3 0 −1−5 2 0 −2

∣∣∣∣∣∣∣∣ d)

∣∣∣∣∣∣∣∣1 −3 5 −22 4 −1 03 2 1 −3−1 −2 3 0

∣∣∣∣∣∣∣∣




Problem 5

Without computing the determinants, show that

a

∣∣∣∣∣∣1 x1 x21 y1 x21 y1 y2

∣∣∣∣∣∣ = (y1 − x1)(y2 − x2)

b

∣∣∣∣∣∣∣∣1 x1 x2 x31 y1 x2 x31 y1 y2 x31 y1 y2 y3

∣∣∣∣∣∣∣∣ = (y1 − x1)(y2 − x2)(y3 − x3)




Problem 6 Answer

Without computing the determinants, find the value of :

a)

∣∣∣∣∣∣∣∣1 1 1 11 1 + a 1 11 1 1 + b 11 1 1 1 + c

∣∣∣∣∣∣∣∣ b)

∣∣∣∣∣∣∣∣1 4 4 44 2 4 44 4 3 44 4 4 4

∣∣∣∣∣∣∣∣ c)

∣∣∣∣∣∣∣∣∣∣1 2 3 4 5−1 0 3 4 5−1 −2 0 4 5−1 −2 −3 0 5−1 −2 −3 −4 0

∣∣∣∣∣∣∣∣∣∣




Problem 7

Without computing the determinants, show that∣∣∣∣∣∣∣∣a 1 1 11 a 1 11 1 a 11 1 1 a

∣∣∣∣∣∣∣∣ = (a+ 3)(a− 1)3




Problem 8 Answer

Given the matrices

A =

1 2 0 00 2 0 00 2 1 00 2 0 1

and B =

1 0 0 30 4 0 40 0 0 −11 1 2 8

Compute:

a) |AB| b)∣∣(BA)t∣∣ c) |2A3B| d) |A+B|




Problem 9 Answer

Let A and B be matrices of order n. Knowing that |A| = 5 and|B| = 3, compute:

a)∣∣BAt

∣∣ b) |3A| c)∣∣(2B)2

∣∣ , B of order 3




Problem 10 Answer

Compute |AB|,∣∣(BA)t∣∣, |2A3B|, knowing that

A =

1 2 0 00 2 0 00 2 1 00 2 0 1

B =

1 0 0 30 4 0 40 0 0 −11 1 2 8




Problem 11 Answer

If A2 = A, what values can |A| have?




Problem 12

Let A be a square matrix of order n such that A2 = −I. Provethat |A| 6= 0 and that n is an even number.




Problem 13

Let A be a square matrix of order n. Prove that AAt is asymmetric matrix.




Problem 14 Answer

Given the following matrices:

A =

(1 11 1

)B =

(0 1−1 1

)C =

(−2 10 0

)solve the matrix equation 3X + 2A = 6B − 4A+ 3C.




Problem 15 Answer

Suppose that A.B and X are matrices of orden n, solve thefollowing matrix equations:

a 3(At − 2B) + 5Xt = −B.b (At +X)t −B = 2A.








1 The matrix A =

4 1 −30 −1 10 0 7

is Answer

a. a lower triangular matrixb. an upper triangular matrixc. a diagonal matrix




2 Which of the following is the true definition of a symmetricmatrix? Answer

a. A square matrix A is said to be symmetric if A = −Ab. A square matrix A is said to be symmetric if A = −At

c. A square matrix A is said to be symmetric if A = At




3 Given A and B matrices of order m× n and n× p, (AB)t

equals to: Answer

a. BtAt

b. AtBt

c. AB




4 Let A be a matrix such that A2 = A then, if B = A− I,then: Answer

a. B2 = Bb. B2 = Ic. B2 = −B




5 Let A and B be square matrices of order 3. If |A| = 3| and|B| = −1 then: Answer

a. |2A � 4B| = (−4)23b. |2A � 4B| = (−3)23c. |2A � 4B| = (−3)29




6 Which of the following properties is NOT always true? Answer

a.∣∣A2∣∣ = |A|2

b. |A+B| = |A|+ |B|c. |AtB| = |A| |B|




7 Given the 3 by 3 matrices A, B, C such that |A| = 2, |B| = 4

and |C| = 3, compute∣∣∣ 1|A|B

tC−1∣∣∣: Answer

a. 23

b.16

c. 6




8 Let A and B be matrices of the same order, which of thefollowing properties is always true? Answer

a. (A−B)(A+B) = A2 −B2

b. (A−B)2 = A2 − 2AB +B2

c. A(A+B) = A2 +AB




9 Which of the following properties is NOT true? Answer

a.∣∣A2∣∣ = |A|2

b. |−A| = |A|c. |At| = |A|




10 Let A and B be symmetric matrices, then which of thefollowing is also a symmetric matrix: Answer

a. BAb. A+Bc. AB




11 Let A be an n by n real matrix. Then, if k ∈ R one has thatAnswer

a. |kA| = k |A|b. |kA| = |k| |A|, being |k| the absolute value of the real numberk

c. |kA| = kn |A|




12 Given the matrices

A =

(1 11 1

)B =

(0 1−1 1

)C =

(−2 10 0

)the solution to the matrix equation3X + 2A = 6B − 4A+ 3C is Answer

a. X =

(1 00 1

)b. X =

(−4 1−4 0

)c. X =

(4 14 1

)




13 Let A,B be matrices of order n. Then: Answer

a. (AB)2= A2B2

b. (AB)2= B2A2

c. (AB)2= A(BA)B




14 Given the matrix A =

1 2 33 1 22 3 1

, the adjoint element a12 is:

Answer

a. −1b. 1c. −2


Answers to Problems

Answers to Problems

Problem 1 Return

a

(−2 10 0

)b 2

(−1/2 −1/21 1

)c

(1 02 0

)d 3

(0 10 1

)e

(2 21 1

)

Answers to Problems

Problem 2 Return

a AB =

(16 1013 −15

)and BA =

12 3 636 8 −4−5 −2 −19

b AB =

−2 1 75 11 53 10 8

and BA =

5 5 48 4 612 5 8

c AB = 3

(2 3

)and BA is not possible

Answers to Problems

Problem 3 Return

It is false because the multiplication of matrices does not verify theconmutative property. To probe its falsity the students mustprovide a counterexample.

Answers to Problems

Problem 4 Return

(a)-24 (b)-58 (c)-80 (d)35

Answers to Problems

Problem 6 Return

(a) abc (b) −24 (c) 5!

Answers to Problems

Problem 8 Return

(a)16 (b) 16 (c) 2834 (d) 130

Answers to Problems

Problem 9 Return

(a)15 (b) 3n · 5 (c) 26 · 32

Answers to Problems

Problem 10 Return

(a)3n · 5 (b) 5n−1 (c) 15

Answers to Problems

Problem 11 Return

|A| = 0 or |A| = 1.

Answers to Problems

Problem 14 Return

X =

(−4 1−4 0

)

Answers to Problems

Problem 15 Return

(a)X = Bt − 35A (b)X = At +Bt.

Answers to Problems

Answers to Multiple choice questions


1 The matrix A =

4 1 −30 −1 10 0 7

is Back

a. a lower triangular matrixb. an upper triangular matrixc. a diagonal matrix


2 Which of the following is the true definition of a symmetricmatrix? Back

a. A square matrix A is said to be symmetric if A = −Ab. A square matrix A is said to be symmetric if A = −At

c. A square matrix A is said to be symmetric if A = At


3 Given A and B matrices of order m× n and n× p, (AB)t

equals to: Back

a. BtAt

b. AtBt

c. AB


4 Let A be a matrix such that A2 = A then, if B = A− I,then: Back

a. B2 = Bb. B2 = Ic. B2 = −B


5 Let A and B be square matrices of order 3. If |A| = 3| and|B| = −1 then: Back

a. |2A � 4B| = (−4)23b. |2A � 4B| = (−3)23c. |2A � 4B| = (−3)29


6 Which of the following properties is NOT always true? Back

a.∣∣A2∣∣ = |A|2

b. |A+B| = |A|+ |B|c. |AtB| = |A| |B|


7 Given the 3 by 3 matrices A, B, C such that |A| = 2, |B| = 4

and |C| = 3, compute∣∣∣ 1|A|B

tC−1∣∣∣: Back

a. 23

b.16

c. 6


8 Let A and B be matrices of the same order, which of thefollowing properties is always true? Back

a. (A−B)(A+B) = A2 −B2

b. (A−B)2 = A2 − 2AB +B2

c. A(A+B) = A2 +AB


9 Which of the following properties is NOT true? Back

a.∣∣A2∣∣ = |A|2

b. |−A| = |A|c. |At| = |A|


10 Let A and B be symmetric matrices, then which of thefollowing is also a symmetric matrix: Back

a. BAb. A+Bc. AB


11 Let A be an n by n real matrix. Then, if k ∈ R one has thatBack

a. |kA| = k |A|b. |kA| = |k| |A|, being |k| the absolute value of the real numberk

c. |kA| = kn |A|


12 Given the matrices

A =

(1 11 1

)B =

(0 1−1 1

)yC =

(−2 10 0

)the solution to the matrix equation3X + 2A = 6B − 4A+ 3C is Back

a. X =

(1 00 1

)b. X =

(−4 1−4 0

)c. X =

(4 14 1

)


13 Let A,B be matrices of order n. Then: Back

a. (AB)2= A2B2

b. (AB)2= B2A2

c. (AB)2= A(BA)B


14 Given the matrix A =

1 2 33 1 22 3 1

, the adjoint element a12 is:

Back

a. −1b. 1c. −2


Quadratic FormsReview problems for Quadratic forms


Quadratic forms

Back

Quadratic forms



Outline

Definiteness of a quadratic form

The sign of a quadratic form attending the principal minors

Quadratic forms with linear constraints

Back

Quadratic forms




Quadratic forms: The general case

Definition 8

A quadratic form in n variables is a function Q of the form

Q(x1, x2, ..., xn) = x′Ax = (x1, x2, ..., xn)

a11 a12 · · · a1n

a21 a22 · · · a2n

......

. . ....

an1 an2 · · · ann

x1

x2

...xn

where x′ = (x1, x2, ..., xn) is a vector and A = (aij)n×n is asymmetric matrix of real numbers.

Then A is called the symmetric matrix associated with Q.

Quadratic forms




Matrix and Polynomial form1 Matrix form of a quadratic form.

Q(x1, x2, ..., xn) =(

x1 x2 · · · xn

)

a11 a12 . . . a1n

a21 a22 · · · a2n

......

. . ....

an1 an2 · · · ann

x1

x2

...xn

2 Polynomial form of a quadratic form. Expanding the matrix

multiplication we obtain a double sum such as

Q(x1, x2, ..., xn) = x′Ax =n∑

i=1

n∑j=1

aijxixj

Function Q is a homogeneous polynomial of degree twowhere each term contains either the square of a variable or aproduct of exactly two of the variables. The terms can begrouped as follows

Q(x) =

n∑i=1

biix2i +

n∑i,j=1,i<j

bijxixj

where bii = aii and, since A is a symmetric matrix withaij = aji ∀i, j = 1, 2, ..., n, where bij = 2aij .Quadratic forms




Q(x1, x2, ..., xn) = (x1, x2, · · · , xn)

a11 a12 . . . a1na21 a22 · · · a2n

.

.

.

.

.

.. . .

.

.

.an1 an2 · · · ann

x1x2

.

.

.xn

=n∑

i=1

biix2i+

n∑i,j=1,i<j

bijxixj

There exits a relationship between the elements in the symmetricmatrix associated with Q and the coefficients of the polynomial.Note that

The elements in the main diagonal of matrix A arethe coefficients of the quadratic terms of thepolynomial.

The elements outside the main diagonal of matrix A(aij = aji i 6= j) are half of the coefficients of thenon quadratic terms of the polynomial.

Quadratic forms




Definition 9

A quadratic form Q(x) = x′Ax (as well as its associatedsymmetric matrix A) is said to be

1 Positive definite if Q(x) > 0 for all x 6= 0.

2 Negative definite if Q(x) < 0 for all x 6= 0

3 Positive semidefinite if Q(x) ≥ 0 for all x 6= 0

4 Negative semidefinite if Q(x) ≤ 0 for all x 6= 0

5 Indefinite if there exist vectors x and y such that Q(x) < 0and Q(y) > 0. Thus, an indefinite quadratic form assumesboth negative and positive values.

Quadratic forms




Definition 10

A principal minor of order r of an n× n matrix A = (aij) is thedeterminant of a matrix obtained by deleting n− r rows and n− rcolumns such that if the ith row (column) is selected, then so isthe ith column (row).

In particular, a principal minor of order r always includes exactly relements of the main (principal) diagonal. Also, if matrix A issymmetric, then so is each matrix whose determinant is a principalminor. The determinant of A itself, |A|, is also a principal minor(No rows or colmns are deleted)

Definition 11

A principal minor is called a leading principal minors of order r(1≤ r ≤ n) if it consists of the first (”leading”) r rows andcolumns of |A|.

Quadratic forms




Theorem

Let Q(x) = x′Ax be a quadratic form of n variables and let

|A1| = a11, |A2| =∣∣∣∣ a11 a12

a21 a22

∣∣∣∣ , |A3| =

∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33

∣∣∣∣∣∣ ..., |An| = |A|

be the leading principal minors of matrix A. Then

Q(x) positive definite ⇔ |A1| > 0, |A2| > 0, ..., |An| > 0Q(x) negative definite ⇔ the leading principal minors of even orderare positive and those of odd order are negative..If |A| = 0 and the remaining leading principal minors are positive⇒ Q is positive semidefinite.If |A| = 0 and the remaining leading principal minors of even orderare positive and those of odd order are negative =⇒ Q is negativesemidefinite.If |A| 6= 0 and the leading principal minors do not behave as in a) orb) ⇒ Q is indefinite.If |A| = 0 and |Ai| 6= 0 i = 1, 2, ..., n− 1 and the leading principalminors do not behave as in c) or d)⇒ Q is indefinite.

Quadratic forms




Theorem

Let Q(x) = x′Ax be a quadratic form of n variables such that|A| = 0. Then

All the principal minors are positive or zero ⇔ Q is positivesemidefinite

All the principal minors are of even order are positive or zeroand those of odd order are negative or zero ⇔ Q is negativesemidefinite.

Quadratic forms




Definition 12

It is said that the quadratic form Q(x) = xtAx is constrained to alinear constraint when

(x1, x2, ..., xn) ∈ {(x1, x2, ..., xn) ∈ Rn/b1x1 + b2x2 + ...+ bnxn = 0}

To find the sign of a constrained quadratic form, follow thefollowing steps:

1 Analyze the sign of Q(x) = xtAx without any constraint. If itis definite (positive or negative), then the constrainedquadratic form is of the same sign.

2 If the unconstrained quadratic form is not definite, we solvethe linear constraint for one variable and substitute it into thequadratic form. The result is an unconstrained quadratic formwith n− 1 variables. We study the sign with the principalminors.

Quadratic forms







Problem 1 Answer

Without computing any principal minor, determine the definitenessof the following quadratic forms:

a Q(x, y, z) = (x + 2y)2 + z2

b Q(x, y, z) = (x + y + z)2 − z2

c Q(x, y, z) = (x− y)2 + 2y2 + z2

d Q(x, y, z) = −(x− y)2 − (y + z)2

e Q(x, y, z) = (x− y)2 + (y − 2z)2 + (x− 2z)2




Problem 2 Answer

Write the following quadratic forms in matrix form with Asymmetric and determine their definiteness.

a Q(x, y, z) = −x2 − y2 − z2 + xy + xz + yz

b Q(x, y, z) = y2 + 2z2 + 2xz + 4yz

c Q(x, y, z) = 2y2 + 4z2 + 2yz

d Q(x, y, z) = −3x2 − 2y2 − 3z2 + 2xz

e Q(x1, x2, x3, x4) = x21 − 4x23 + 5x24 + 4x1x3 + 2x2x3 + 2x2x4




Problem 3 Answer

Investigate the definiteness of the following quadratic formsdepending on the value of parameter a.

a Q(x, y, z) = −5x2 − 2y2 + az2 + 4xy + 2xz + 4yz

b Q(x, y, z) = 2x2 + ay2 + z2 + 2xy + 2xz

c Q(x, y, z) = x2 + ay2 + 2z2 + 2axy + 2xz




Problem 4 Answer

Find the value of a which makes the quadratic formQ(x, y, z) = ax2 + 2y2 + z2 + 2xy + 2xz + 2yz be semidefinite. Forsuch a value, determine its definiteness when it is subject tox− y − z = 0?




Problem 5 Answer

If A =

0 0 −10 1 −2−1 −2 3

then

a investigate its definiteness.

b Write the polynomial and matrix forms of the quadratic formQ(h1,h2, h3) which is associated with matrix A.

c determine its definiteness when it is subject toh1 + 2h2 − h3 = 0.




Problem 6 Answer

Investigate the definiteness of the following matrices. Write thepolynomial and matrix form of the associated quadratic forms:

A =

−2 1 11 −2 11 1 −2

B =

2 −3/2 1/2−3/2 1 −1/21/2 −1/2 0

C =

1 −1 1−1 1 −11 −1 3

D =

−5 2 1 02 −2 0 21 0 −1 00 2 0 −5




Problem 7 Answer

Determine the definiteness of the following constrained quadraticforms.

a Q(x, y, z) = (x, y, z)

−2 1 11 −2 11 1 −2

xyz

s. t x + y − 2z = 0

b Q(x, y, z) = (x, y, z)

2 −3/2 1/2−3/2 1 −1/21/2 −1/2 0

xyz

s. t x− y = 0

c Q(x, y, z) = (x, y, z)

1 −1 1−1 1 −11 −1 3

xyz

s. t x + 2y − z = 0

d Q(x1, x2, x3, x4) = (x1, x2, x3, x4)

−5 2 1 02 −2 0 21 0 −1 00 2 0 −5

x1x2x3x4

s. t

2x1 − 4x4 = 0




Problem 8 Answer

Let Q(x, y, z) = −x2 − 2y2 − z2 + 2xy − 2yz be a quadratic form

a write its matrix form and investigate its definiteness.

b Investigate its definiteness if it is constrained to2x− 2y + az = 0 for the different values parameter a canhave.




Problem 9 Answer

Determine the definiteness of the Hessian matrix of the followingfunctions

a f(x, y, z) = 2x2 + y2 − 2xy + xz− yz + 2x− y + 8

b f(x, y) = x4 + y4 + x2 + y2 + 2xy

c f(x, y, z) = ln(x) + ln(y) + ln(z)




Problem 10 Answer

Determine the definiteness of the Hessian matrix of the followingproduction functions when K,L > 0.

a) Q(K,L) = K1/2L1/2 b) Q(K,L) = K1/2L2/3




Problem 11 Answer

The production function Q(x, y, z)=ax2 + 4ay2 + a2z2−4axy, witha > 0, relates the produced quantity of a good to three rawmaterials (x, y and z) used in the production precess.

a Determine the definiteness of Q(x, y, z).

b Knowing that if x = y = z = 1 then six units of a good areproduced, find the value of parameter a.

c Using the value of a found in (b), determine the definitenessof Q(x, y, z) when the raw materials x and y are used in thesame quantity.








1 Which of the following is a quadratic form? Answer

a Q(x, y, z) = x2 + 3z2 + 6xy + 2zb Q(x, y, z) = 2xy2 + 3z2 + 6xyc Q(x, y, z) = 3xy + 3xz + 6yz




2 Let Q(x, y, z) be a quadratic form such that Q(1, 1, 0) = 2and Q(5, 0, 0) = 0, then

Answer

a Q(x, y, z) could be indefiniteb Q(x, y, z) is positive definitec Q(x, y, z) could be negative semidefinite




3 The quadratic form Q(x, y, z) = (x− y)2 + 3z2 isAnswer

a positive definiteb positive semidefinitec indefinite




4 The quadratic form Q(x, y, z) = −y2 − 2z2 isAnswer

a negative definiteb negative semidefinitec indefinite




5 The quadratic form Q(x, y, z) = x2 + 2xz + 2y2 − z2 isAnswer





6 A 2 by 2 matrix has a negative determinant, then the matrix isAnswer





7 The matrix A =

1 1 01 1 00 0 2

is

Answer

a indefiniteb positive semidefinitec positive definite




8 The leading principal minors of a 4 by 4 matrix are|A1| = −1, |A2| = 1, |A3| = 2, and |A4| = |A| = 0.Then,

Answer

a the matrix is negative semidefiniteb its definiteness cannot be determined with this informationc the matrix is indefinite




9 The leading principal minors of a 4 by 4 matrix are|A1| = −1, |A2| = 1, |A3| = −2 and |A4| = |A| = 0.Then,

Answer

a the matrix is negative semidefiniteb its definiteness cannot be determined from this informationc the matrix is indefinite




10 The leading principal minors of a 4 by 4 matrix are|A1| = −1, |A2| = 1, |A3| = −2, and |A4| = |A| = 1.Then,the matrix is

Answer

a negative definiteb indefinitec positive definite and negative definite




11 The quadratic form in three variables Q(x, y, z), subject tox+ 2y − z = 0, is positive semidefinite. Then, theunconstrained quadratic form is:

Answer

a positive semidefinite or indefiniteb positive semidefinitec positive definite or positive semidefinite




12 If Q(x, y, z) is a negative semidefinite quadratic form suchthat Q(−1, 1, 1) = 0, then Q(x, y, z) subject to the constraintx+ 2y − z = 0

Answer

a is negative semidefiniteb cannot be classified with this informationc is negative definite or negative semidefinite




13 The quadratic form Q(x, y, z) = (x− y)2 + 3z2 subject to theconstraint x = 0 is:

Answer





14 The quadratic form Q(x, y, z) = (x− y)2 + 3z2 subject to theconstraint z = 0 is:

Answer



Solutions

Answers to the problems



Problem 1 Return

a positive semidefinite.

b indefinite.

c positive definite.

d negative semidefinite.

e positive semidefinite.


Problem 2 Return

(a) Q(x, y, z) = (x, y, z)

−1 1/2 1/21/2 −1 1/21/2 1/2 −1

xyz

negative semidefinite.

(b) Q(x, y, z) = (x, y, z)

0 0 10 1 21 2 2

xyz

indefinite.

(c) Q(x, y, z) = (x, y, z)

0 0 00 2 10 1 4

xyz

positive semidefinite.

(d) Q(x, y, z) = (x, y, z)

−3 0 10 −2 01 0 −3

xyz

negative definite.

(e) Q(x1, x2, x3, x4) = (x1, x2, x3, x4)

1 0 2 00 0 1 12 1 −4 00 1 0 5

x1x2x3x4

indefinite.


Problem 3 Return

(a) negative definite for a < −5, negative semidefinite for a = −5and indefinite when a > −5.(b) positive definite if a < 1, positive semidefinite if a = 1 andindefinite when a < 1.(c) Indefinite when a < 0 or a > 1/2, positive semidefinite fora = 0 or a = 1/2 and positive definite.if 0 < a < 1/2.


Problem 4 Return

a = 1. The constrained quadratic form is positive definite.


Problem 5 Return

(a) indefinite.

(b) Q(h1, h2,h3) = (h1, h2,h3)

0 0 −10 1 −2−1 −2 3

h1h2h3

=

h22 + 3h23 − 2h1h3 − 4h2h3(c) positive definite.


Problem 6 Return

(a) negative semidefinite,Q(x, y, z) = −2x2 − 2y2 − 2z2 + 2xy + 2xz + 2yz.(b) indefinite, Q(x, y, z) = 2x2 + y2 − 3xy + xz− yz.(c) positive semidefinite,Q(x, y, z) = x2 + y2 + 3z2 − 2xy + 2xz− 2yz.(d) negative definite,Q(x1, x2, x3, x4) =−5x21 − 2x22 − x23 − 5x24 + 4x1x2 + 2x1x3 + 4x2x4.


Problem 7 Return

(a) negative semidefinite. (b) The constrained quadratic form isnull.(c) positive definite. (d) negative definite (since the unconstrainedquadratic form is negative definite).


Problem 8 Return

(a) Q(x, y, z) = (x, y, z)

−1 1 01 −2 −10 −1 −1

xyz

negative

semidefinite.(b) negative semidefinite if a = 0, negative definite if a 6= 0.


Problem 9 Return

(a) Hf(x, y, z) is indefinite∀(x, y, z) ∈ R3.(b) Hf(x, y) is positive definite∀(x, y) ∈ R2 − {(0, 0)}. H f(0,0) ispositive semidefinite.(c) Hf(x, y, z) is negativedefinite∀(x, y, z) ∈ Dom(f) = {(x, y, z) ∈ R3/x > 0, y > 0, z > 0}.


Problem 10 Return

(a) negative semidefinite. (b) indefinite.


Problem 11 Return

(a) Positive Semidefinite.(b) a = 2.(c) Positive Definite.




1 Which of the following is a quadratic form? Back

a Q(x, y, z) = x2 + 3z2 + 6xy + 2zb Q(x, y, z) = 2xy2 + 3z2 + 6xyc Q(x, y, z) = 3xy + 3xz + 6yz


2 Let Q(x, y, z) be a quadratic form such that Q(1, 1, 0) = 2and Q(5, 0, 0) = 0, then

Back

a Q(x, y, z) could be indefiniteb Q(x, y, z) is positive definitec Q(x, y, z) could be negative semidefinite


3 The quadratic form Q(x, y, z) = (x− y)2 + 3z2 isBack



4 The quadratic form Q(x, y, z) = −y2 − 2z2 isBack



5 The quadratic form Q(x, y, z) = x2 + 2xz + 2y2 − z2 isBack



6 A 2 by 2 matrix has a negative determinant, then the matrix isBack



7 The matrix A =

1 1 01 1 00 0 2

is

Back



8 The leading principal minors of a 4 by 4 matrix are|A1| = −1, |A2| = 1, |A3| = 2, and |A4| = |A| = 0.Then,

Back

a the matrix is negative semidefiniteb its definiteness cannot be determined with this informationc the matrix is indefinite


9 The leading principal minors of a 4 by 4 matrix are|A1| = −1, |A2| = 1, |A3| = −2 and |A4| = |A| = 0.Then,

Back

a the matrix is negative semidefiniteb its definiteness cannot be determined from this informationc the matrix is indefinite


10 The leading principal minors of a 4 by 4 matrix are|A1| = −1, |A2| = 1, |A3| = −2, and |A4| = |A| = 1.Then,the matrix is

Back

a negative definiteb indefinitec positive definite and negative definite


11 The quadratic form in three variables Q(x, y, z), subject tox+ 2y − z = 0, is positive semidefinite. Then, theunconstrained quadratic form is:

Back

a positive semidefinite or indefiniteb positive semidefinitec positive definite or positive semidefinite


12 If Q(x, y, z) is a negative semidefinite quadratic form suchthat Q(−1, 1, 1) = 0, then Q(x, y, z) subject to the constraintx+ 2y − z = 0

Back

a is negative semidefiniteb cannot be classified with this informationc is negative definite or negative semidefinite


13 The quadratic form Q(x, y, z) = (x− y)2 + 3z2 subject to theconstraint x = 0 is:

Back



14 The quadratic form Q(x, y, z) = (x− y)2 + 3z2 subject to theconstraint z = 0 is:

Back



Useful links

Links to the Wolfram Demostrations Project web page

Matrix Multiplication >>

Matrix Transposition >>

Determinants by expansion >>

Determinants using diagonals >>

Bibliography

Matrix algebra:

Sydsaeter,K. and Hammond,P.J. Essential Mathematics for EconomicAnalysis. Prentice Hall. New Jersey. Pages: 537-554 and 573-591. >>

Quadratic forms

Sydsaeter,K. and Hammond,P.J. Further Mathematics for EconomicAnalysis. Prentice Hall. New Jersey. Pages: 28-37. >>

Back


http://demonstrations.wolfram.com/MatrixMultiplication/

http://demonstrations.wolfram.com/MatrixTransposition/

http://demonstrations.wolfram.com/33DeterminantsByExpansion/

http://demonstrations.wolfram.com/33DeterminantsUsingDiagonals/

http://books.google.es/books?id=ZliZosMpub8C&printsec=frontcover&hl=es&source=gbs_ge_summary_r&cad=0

http://books.google.es/books?id=mqgK08S94NAC&printsec=frontcover&hl=es&source=gbs_ge_summary_r&cad=0

Documents

Mathematics for Business Administration: Matrix Algebra